Cant apply texture to object because it is missing UV coordinates
Since Blender 2.8, curves have a UV map all the time.
If using versions before 2.80, as mentioned in this post, you need to enable the Use UV for mapping option in Properties Editor &gt; Object Data Tab &gt; Texture space panel:

Alembic export with UVs enabled in export options should work just fine.
I wasn't able to test on your scene since the curve object was empty. But theoretically it should work.